Trigonometric approximation and a general form of the Erdos Turan inequality

There exists a positive function psi(t) on t >= 0 with fast decay infinity such that for every measurable set Omega in the Euclidean space and R > 0 there exist entire functions A (x) and B (x) of exponential type R satisfying A(x) <= (chi Omega)(x) <= B(x) and |B(x) - A(x)| <= psi(R dist (x, boundary (Omega))). This leads to Erdos Turan estimates for discrepancy of point set distributions in the multi-dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.